UNIVERSITY 


THE  ACTION 


JUPITER  UPON  COMET  V,  1889, 


CHARLES   LANE   POOR,    M.  S. 


A  THESIS  SUBMITTED  TO  THE  JOHNS  HOPKIXS  UNIVERSITY  FOR 

THE   DEGREE   OP   DOCTOR   OF   PHILOSOPHY. 


,    1891. 


TIVRH? 
I  *  v  f 


BALTIMORE  : 

JOHN  MURPHY  &  CO. 
1892. 


THE    ACTION 


OF 


JUPITER  UPON  COMET  V,  1889, 


BY 


CHARLES  LANE  POOR,   M.  S. 


A  THESIS  SUBMITTED  TO  THE  JOHNS  HOPKINS  UNIVERSITY  FOR 

THE   DEGREE   OF   DOCTOR  OF   PHILOSOPHY. 


,    1891. 


BALTIMORE : 

JOHN   MURPHY  &  CO. 
1892. 


INTRODUCTION. 


On  July  6th,  1889,  at  Geneva,  New  York,  Brooks  discovered  a  faint 
telescopic  comet,  since  known  as  Comet  d  and  V  1889.  During  the  fol- 
lowing summer  and  early  fall  few  observations  were  made.  Soon,  however, 
the  periodic  character  of  this  body  was  recognized,  and  when  Mr.  Searle 
pointed  out  that  in  1886  it  must  have  passed  very  close  to  Jupiter  the 
interest  of  astronomers  was  at  once  aroused.  Mr.  Chandler  of  Boston 
verified  this  suggestion,  and  undertook  the  determination  of  the  effect 
of  this  close  approach.  His  results  were  published  in  the  Astronomical 
Journal,  No.  205,  where  he  showed  that  the  orbit  had  been  radically 
changed,  that  before  1886  the  comet  was  moving  in  an  entirely  differ- 
ent ellipse  from  that  in  which  it  is  at  present  moving.  Before  any  note 
on  the  subject  had  been  published,  Prof.  Newcomb,  being  unaware  that 
any  one  was  then  working  on  the  problem,  suggested  it  as  an  appro- 
priate subject  for  a  thesis.  I  undertook  the  work,  made  a  series  of 
observations,  deduced  new  elements,  and  was  busy  computing  the  per- 
turbations, when  Mr.  Chandler's  first  paper  appeared.  I  at  once  laid 
the  work  aside,  but  after  some  months,  with  Mr.  Chandler's  assent,  I 
again  attacked  the  very  fascinating  problem. 

At  present  the  comet  is  moving  in  a  small  seven  years'  ellipse. 
Mr.  Chandler  found  that  before  the  encounter  with  Jupiter  the  comet 
had  been  moving  in  a  large  ellipse,  with  a  twenty-seven  years'  period, 
whose  perihelion  was  almost  exactly  the  present  aphelion  distance.  That 
is,  the  original  orbit  was  not  only  much  larger  than  the  present  one, 
but  its  position  in  space  was  different,  the  directions  of  the  lines  of 
apsides  and  nodes  were  nearly  completely  reversed,  and  the  plane  of 
the  orbit  was  tilted  a  number  of  degrees.  Again  Mr.  Chandler  shows 
that,  assuming  the  above  changes  to  be  substantially  correct,  there  could 
have  been  no  previous  close  approach  to  any  planet,  sufficient  to  appre- 

3 


\ 59872 


4  POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 

ciably  disturb  the  comet's  orbit,  back  to  the  year  1779.  But  that  in 
that  year  the  comet  again  passed  under  the  control  of  Jupiter  and  then 
experienced  a  radical  change  of  orbit,  and  that  at  the  same  point  of 
longitude  when  Lexell's  comet  underwent  its  notable  disturbance  in  that 
year.  Moreover  the  most  probable  elements  of  Lexell's  comet  subse- 
quent to  its  disturbance  and  those  of  Comet  V  previous  to  1886  show  such 
a  striking  likeness,  that  one  at  once  infers  the  identity  of  the  two  bodies. 

Lexell's  comet,  it  will  be  remembered,  was  discovered  by  Messier 
on  the  night  of  June  14-15,  1770.  He  at  first  took  it  to  be  a  small 
nebula,  and  did  not  recognize  its  cometary  character  until  two  or  three 
nights  later.  The  comet  was  then  rapidly  approaching  the  earth ;  it 
became  visible  to  the  naked  eye  on  June  21st,  and  on  July  2nd,  passed 
closer  to  the  earth  than  any  other  known  comet.  It  was  then  about 
as  bright  as  the  North  Star,  and  seen  through  a  telescope,  its  diam- 
eter was  found  to  be  about  twice  that  of  full  moon,  and  it  had  a  well 
marked,  though  small  tail.  On  July  4th,  it  passed  into  the  rays  of 
the  sun,  and  became  invisible,  only  to  reappear,  however,  on  August 
4th,  and  to  remain  visible  to  the  naked  eye  until  the  26th,  and  to  the 
telescope  until  October  2nd. 

Lexell  was  the  first  to  point  out  tha.t  this  comet  was  periodic,  was 
then  revolving  about  the  sun  in  an  ellipse  of  5.58  years.  To  the  objec- 
tion that  it  had  not  been  seen  six  years  before,  he  proved  that  in  1767, 
it  had  been  in  conjunction  with  Jupiter,  and  that  as  their  mutual  dis- 
tance was  only  *!*  that  of  the  comet  from  the  sun,  the  action  of  Jupiter 
had  probably  altered  its  orbit  considerably.  He  also  predicted  a  second 
close  approach  in  1779,  and  said  that  this  might  prevent  its  reappear- 
ance after  that  date.  This  prediction  of  Lexell's  was  fulfilled,  for  the 
comet  was  never  again  seen,  unless,  indeed,  it  prove  that  the  comet 
discovered  by  Brooks  on  July  6,  1889,  is  this  lost  body. 

About  1845,  Le  Verrier  carefully  and  completely  worked  out  the 
theory  of  this  comet's  movements.  His  conclusions  were  substantially  the 
same  as  Lexell's.  The  main  fact  was  perfectly  clear;  the  comet  had 
passed  very  close  to  Jupiter  in  1779,  perhaps  had  even  passed  in  among 
the  satellites  of  Jupiter.  The  resulting  changes  in  the  orbit  were  enor- 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889.  5 

mous.  l3ut  unfortunately  it  was  found  that  the  original  observations 
were  not  exact  enough  to  completely  determine  the  orbit  of  the  comet — 
various  systems  of  elements  would  equally  well  represent  the  observations. 
The  differences  were  small,  but  became  magnified  many  times  in  the 
various  orbits  deduced  for  the  comet  after  its  appulse  with  Jupiter.  Le 
Verrier  expressed  these  different  sets  of  elements  in  terms  of  an  indeter- 
minate quantity  (i ;  the  unit  of  which  corresponds  to  an  arbitrary  change 
of  0.01  in  the  semi-major  axis  of  the  orbit  that  the  comet  had  in  1770. 
He  gives  a  table  containing  the  resulting  values  of  the  elements  for  dif- 
ferent assumed  values  of  p.  On  the  hypothesis  that  fi  =  0  the  comet 
passed  the  centre  of  Jupiter  at  no  greater  distance  than  three  and  a  half 
radii  of  that  planet ;  the  resulting  orbit  in  this  case  would  be  an  hyper- 
bola and  the  comet  would  have  vanished  forever.  Again,  on  the  most 
probable  hypothesis  of  fi  —  1  according  to  Le  Verrier,  the  comet  passed 
Jupiter  at  a  distance  of  one  hundred  of  that  planet's  radii,  and  the  result- 
ing orbit  was  an  ellipse  of  about  nine  years'  period.  Besides  these  extreme 
cases  we  have  for  different  values  of  ft  an  immense  variety  of  resulting 
orbits.  Such  were  Le  Verrier's  conclusions: — the  original  data  furnished 
no  complete  solution  to  the  problem  of  what  became  of  Lexell's  comet 
after  its  encounter  with  Jupiter  in  1779. 

For  the  value  of  ^  =  0.35  the  resulting  elements  of  Lexell's  comet 
agree  very  well  with  those  derived  by  Mr.  Chandler  for  Comet  V  before 
its  disturbance  in  1886.  Yet  this  agreement  is  by  no  means  close  enough 
to  establish  the  identity  of  the  two  bodies.  The  presumption,  however, 
in  favor  of  such  identity  is  very  strong,  -as  Mr.  Chandler  clearly  points 
out.  Again  it  must  be  remembered  that  his  numerical  results  are  but  a 
first  approximation.  He  takes  account  only  of  the  principal  perturbations, 
that  is,  the  action  of  Jupiter  is  only  considered  during  the  few  months  of 
very  close  approach.  With  his  assent  I  have  carried  his  work  a  few 
steps  farther,  making  a  second  approximation  to  the  numerical  solution 
of  the  problem. 

While  the  numerical  part  of  my  work  is  to  be  regarded  only  as  a 
second  step,  I  have  tried  to  make  the  methods  pursued  as  complete  as 
possible.  Where  new  or  rare  formulae  are  used  the  derivation  of  them 


6  POOR,  Tlie  Action  of  Jupiter  upon  Comet  V,  1889. 

is  shown  and  their  use  clearly  explained.  The  numerical  computations 
were  made  during  the  summer  of  1890  and  the  results  published,  as  a 
note,  in  Astronomical  Journal,  No.  228.  Before  the  thesis  was  finished, 
however,  some  unexpected  and  very  valuable  observations  of  the  comet 
were  obtained  at  the  Lick  Observatory  during  the  months  of  November 
and  December,  1890,  or  nearly  nine  months  after  the  regular  series  ended. 
These  observations  have  been  inserted  into  the  original  manuscript,  and 
although  the  resulting  changes  are  not  very  marked,  yet  they  caused, 
practically,  an  entire  re-computation  of  the  numerical  part  of  this  work. 
The  results  thus  obtained  appear  in  the  following  pages  and  were  pub- 
lished in  Astronomical  Journal,  No.  244.  They  give,  I  think,  a  very  good 
approximation  to  the  definitive  determination  of  the  elements  and  of  the 
character  of  the  approach  to  Jupiter  in  1886. 

Note. — February,  1892. — I  am  now  engaged  in  computing  definitive  elements  for  this  interesting  body. 
When  that  work  is  completed  I  shall  carry  the  elements  back  to  1886  and  determine  as  accurately  as  possible 
all  the  phenomena  of  the  approach  in  that  year. 

The  entire  work  may  be  conveniently  divided  into  the  following  parts  : 

FIRST.  The  derivation  of  a  set  of  elements  that  represent  the  motion 
of  the  comet  about  the  sun  at  the  moment  when  I  chose  to  transpose 
Jupiter  and  the  sun,  as  central  force  and  disturbing  force,  respectively. 
This  part  includes  the  correction  of  a  preliminary  orbit  by  comparison 
with  the  observations,  and  a  computation  of  the  principal  perturbations 
suffered  by  the  comet  between  the  time  of  quitting  Jupiter  in  1886,  till 
the  time  of  appearance  in  1889.  There  is  in  this  section  nothing  but 
ordinary  routine  astronomical  work. 

SECOND.  The  transformation  of  the  centre  of  motion  from  the  sun 
to  Jupiter,  and  a  computation  of  the  solar  perturbations  during  the  time 
that  Jupiter  is  regarded  as  the  central  force.  This  section  contains  the 
main  part  of  the  mathematical  ti'eatment  of  the  problem,  and  the  deri- 
vation and  discussion  of  the  various  formulae  needed. 


POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889.  7 

THIRD.  The  transformation  from  Jupiter  to  the  sun  as  the  central 
force,  and  the  computation  of  the  perturbations  by  Jupiter  for  a  few 
months  before  the  appulse.  This  section  is  similar  in  character  to  the 
second,  but  the  mathematical  treatment  is  less  extensive. 

FOURTH.  Discussion  of  the  results  and  a  comparison  of  the  final  orbit 
obtained  with  Le  Verrier's  determination  of  the  orbit  of  Lexell's  'comet. 


SECTION  FIRST. 


CORRECTION  OF  THE  PRELIMINARY  ORBIT  AND  COMPUTATION 
OF  THE  PERTURBATIONS. 

I.     Correction  of  the  approximate  elements  of  the  comet's  orbit. 

The  elements  given  by  Mr.  Chandler  in  the  Astronomical  Journal,  No. 
205,  were  used  as  the  approximate  elements  to  be  corrected  by  the  method 
of  the  "  Variation  of  two  Geocentric  Distances."  These  elements  are : 

T-  1889,  Sept.  30.0119  Gr.  M.  T. 

n=  1°  26'  17.3"! 
fl  =  17    58   45.3   M  890.0 

i=  6     4   10.5  J 

e-  0.470704 

a=  3.684682 

q-  1.950229 

Period  7.0730  years. 

With  the  above  elements,  a  partial  ephemeris  was  computed  and  ten 
normal  places  formed.  Each  place  was  formed  from  the  observations 
made  on  not  more  than  three  days  and  observations  from  the  large 
observatories  only  were  used.  The  places  were  corrected  for  aberration 
and  parallax  with  the  following  results : 

BIGHT   ASCENSION.  DECLINATION.  WEIGHT.  OBSERVATORY. 

Mt.  H. 

Munich  &  Hamburg. 
Washington. 
Wash.  &  Balto. 
Princeton. 
Washington. 


1889. 

h.   m.    s. 

O     1       II 

July  9.5 

23  46  59.47 

8  52  34.0 

3 

Aug.  1.5 

0  4  26.61 

7  0  27.1 

3 

Sept.  27.5 

23  49  42.40 

5  12  13.0 

2 

Oct.  18.5 

23  40  44.45 

3  54  34.5 

3 

Nov.  15.5 

23  47  49.95 

0  40  11.7 

3 

Dec.  22.5 

0  27  4.95 

+  5  29  12.9 

3 

8 

POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889.  9 

RIGHT  ASCENSION.  DECLINATION.  WEIGHT.  OBSERVATORY. 

1890.  h.      m.  8.  °        ' 

Jan.  13.5  1     0  11.94  -f    9  35  46.5  3  .    Princeton. 

Feb.  14.5  1  55  20.32  + 15  27  43.9  3  Princeton. 

Nov.  22.0  9    0  34.28  +  24  13  13.1  1  Mt.  H. 

Dec.  21.0  8  54  10.20  +  25  25  1.70  1  Mt.  H. 

The  above  right  ascensions  and  declinations,  which  refer  to  the 
apparent  equinox  and  equator,  were  then  reduced  to  the  mean  equinox 
and  equator  of  the  beginning  of  the  years  1889  and  1890  respectively. 

Assuming  as  correct,  log  p  =  0.1560744  and  log  p'=  0.3922000  which 
are  the  values  of  the  geocentric  distances  for  July  9.5  and  Feb.  14.5 
respectively,  I  have,  for  the  complete  geocentric  positions  of  the  comet  on 

these  dates : 

July  9.5  Feb.  14.5 

a  =  -  3°  15'  16.8"  )  a  =  +  28°  fifr  14.4"  ) 

«  =  -8    52   38.1   J1  a  =  +  16    27   47.2  P 

log  p  =  0.1560744  log  p'=  0.3922000 

Transforming  these  geocentric  positions  into  the  corresponding  helio- 
centric positions  and  then  referring  the  second  place  to  the  mean  equi- 
nox and  equator  of  1889.0,  I  have : 

July  9.5  Feb.  14.5 

a,  =  -  33°  12'  29.5"  2,  =  +  55°  52'  57.7" 

£  =  -444     4.0  £  =  +344   21.2 

log  r  =  0.3157274  log  f  =  0.3528720 

From  these  a  set  of  elements  was  computed  which  exactly  represents 
the  two  observed  places.  These  elements  are  elsewhere  designated  as 
belonging  to  hypothesis  (°).  An  increment  Ap  =  —  0.001,  a  change  of 
unity  in  the  third  place  of  the  logarithm,  was  then  assigned  to  p,  and 
with  the  geocentric  distances  p  +  Ap  and  p',  was  computed  a  second  set 
of  elements  designated  as  (').  Xext,  a  similar  increment  Ap'=:  — 0.001, 
was  assigned  to  p',  and  from  p  and  p'+  Ap'  a  third  set  of  elements,  desig- 
nated as  ("),  computed.  These  sets  of  elements,  each  of  which  exactly 
represents  the  two  normal  places  of  July  9.5  and  Feb.  14.5,  are: 
2 


10 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 


7t  1°  28'     7".24 

ft  17    57   17.46 

i  6      4     9.00 

log  e  9.6735694 

"    a  0.5670766 

"    u  2.6993917 


1°  16'    7".62 
17    54  53.39 
6      3   49.60 
9.6724582 
0.5657482 
2.7013843 


T  1889  Sept.  30.08039        Sept.  29.603867 


1°  43'  27".88 
17    59   24.92 
6      3  58.23 
9.6708118 
0.5643681 
2.7034544 
Sept.  30.73025 


The  right  ascension  and  declination  for  the  date  of  each  intermediate 
normal  were  then  computed  from  each  set  of  elements.  The  results,  as 
well  as  the  observed  right  ascension  and  declination,  are  exhibited  in  the 
following  table  ;  where  they  are  reduced  to  the  mean  equator  and  equinox 
of  1889  and  1890  respectively  : 


1889. 
Aug.    1.5 

Sept.  27.5 
Oct.  18.5 
Nov.  15.5 
Dec.  22.5 

1890. 
Jan.  13.5 

Nov.  22.0 
Dec.  21.0 


OBSERVED, 
h.       ID.  8. 

a       04  23.79 
3—7°  0'32".5 

23  49  41.25 
5  12  24.5 

23  40  43.10 
3  54  43.3 

23  47  48.45 
-   0  40  22.0 

0  27     2.98 
+   5  29    0.3 

1  0  12.94 
+    9  35  52.2 

9    0  32.35 
+  24  13  19.2 

8  54    7.78 
+  25  25    9.7 


HYP.   °. 
h.      m.          s. 

0    4  25.05 

7°  0'16".4 

23  49  44.87 
5  11  23.0 

23  40  48.73 
3  53  54.58 

23  47  53.24 
-   0  39  28.2 

0  27     6.56 
+    5  29  37.6 


HYP.   '. 
h.      in.          s. 

0    4  27.11 

—    6°59'57".2 

23  49  45.84 
5  10  37.4 

23  40  49.31 
3  53    3.2 

23  47  54.34 
0  38  46.0 

0  27     7.94 
+    5  30    4.9 


HYP.    ". 

h.      m.  8. 

0    4  22.09 
7°  0'51".4 

23  49  31.78 
5  13  40.40 

23  40  33.23 
3  56  19.7 

23  47  37.93 
0  41  45.1 

0  26  55.69 
+    5  28  10.2 


1    0  14.68          1    0  15.71  1    0    7.73 

+    9  36    9.1  +9  36  24.80  +    9  35  17.6 

8  59  58.43          8  59  52.85  9     1  43.08 

+  24  15  44.70  +  24  15  50.50  +  24    7  29.80 

8  53  30.07          8  53  23.54  8  55  32.16 

+  25  28  10.0  +25  28  16.6  +25  19  17.3 


POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889. 
Then  for  each  intermediate  normal  we  have: 

v  A  «•  da    .  «.da 

cos  dAa  =  cos  d  -r-  Ap  +  cos  d  -T-.  Ap 
dp  dp' 


11 


"= 


dS 
dp 


-,Ap' 


Where, 


a-a 


~ 
dp 


=  a"  -  a° 
dp' 


-,  _,  _ 

dp  dp' 

Whence  we  have  'the  equations  of  condition  to  determine  Ap  and 


FOR   RIGHT   ASCENSION. 


WEIGHT. 


Aug.  1.5 
Sept.  27.5 
Oct.  18.5 
Nov.  15.5 
Dec.  22.5 
Jan.  13.5 
Nov.  22.0 
Dec.  21.0 

Aug.  1.5 
Sept.  27.5 
Oct.  18.5 
Nov.  15.5 
Dec.  22.5 
Jan.  13.5 
Nov.  22.0 
Dec.  21.0 


-  18".90  =:  +30".90  Ap-    44".40  Ap' 

-   0".5 

3 

—  54.30  =  +14.55 

—   196.35 

+   8.6 

2 

-   84.45  =  +   8.70 

—  232.50 

1.1 

3 

-   71.  85  =  +  16.50 

-   229.65 

+   2.8 

3 

53.70  =  +  20.70 

163.05 

0.1 

3 

27.  60  =  +  15.45 

104.25 

+   7.0 

3 

+  508.80  =  -86.20 

+  1569.50 

+   9.8 

1 

+  565.65  =  -98.07 

+  1831.25 

-  6.3 

1 

FOR    DECLINATION. 

16".10  =  + 19".20  Ap- 

61.50  =  +  45.60 

48.70  =  +  51.40 

53.80  =  +42.20 

37.30  =  +  27 .60 

16.90  =  +  15.70 

145.46  =  +  5.80 

180.30  =  +   6.60 


RESIDUALS. 


WEIGHT. 


35"  .00  Ap' 

—  2".4 

3 

137.40 

-11.0 

2 

145.10 

+  4.1 

3 

136.90 

-  5.0 

3 

87.40 

-  6.0 

3 

51.56 

+  1.3 

3 

494.90 

+  9.2 

1 

532.70 

-14.3 

1 

Solving  these  equations  by  the  method  of  least  squares  : 

Ap  =  — 0.150831 
Ap'  =  + 0.309693 


12  POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889. 

in  negative  units  of  the  third  decimal  of  the  logarithm.     Hence, 

log  p  =  0.1562252 
log  p'  =  0.3918903 

and  the  complete  geocentric  positions  become, 

July  9.5  Feb.  14.5 

a  =  -  3°  15'  16-.8  )  +28°  50'  14".4  1 

*  =  -8    52  31.1  J3  +15    27   47.2   P 

log  p  =  0.1562252  log  p'  =  0.3918903 

From  these  corrected  geocentric  positions  the  final  elements  were  com- 
puted, and  the  positions  for  the  intermediate  dates,  as  derived  from  them, 
compared  with  the  original  normal  places.  The  resulting  residuals  (ob- 
served— computed),  while  not  as  small  as  could  be  wished  for,  still 
show  that  the  elements  are  exact  enough  for  my  purpose. 

These  elements  are : 

n=   1°  35'  3l".53^ 
H  =  17    59  32.97    [  1890.0 
i=   6      4   13.18  J 
log  a  =  0.5664512 
"    e  —  9.6729017 
"    ft  =  2.7003308 

T—  1889  Sept.  30.355026  Gr.  M.  T. 

II.     Perturbations  by  Jupiter  from  October  1886  to  July  1889. 

Considering  the  above  elements  as  osculatory  for  July  2nd,  1889,  I 
computed  the  perturbations  by  Jupiter  from  that  date  to  the  time  of 
appulse  in  October  1886.  The  ordinary  method  of  the  "  Variation  of  Con- 
stants "  as  given  by  Oppolzer  was  followed.  Until  March  15th,  1887,  an 
interval  of  forty  days  was  used.  At  this  date  the  perturbations  were 
integrated  and  applied  to  the  elements.  With  the  osculating  elements 
thus  derived  for  March  15th,  1887,  the  perturbations  were  computed 
until  October  26th,  1886,  using  an  interval  of  ten  days,  when  they  were 
again  integrated  and  applied  to  the  elements.  In  order  to  take  account 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 


13 


of  quantities  of  the  second  order  in  computing  the  differential  coefficients, 
the  variable  elements  were  used  in  the  computation  for  the  various  dates 
between  March  1887  and  October  1886. 

The  values  of  the  differential  coefficients  and  their  summation  are 
tabulated  for  the  various  elements.  These  tables  are  inserted  at  the  end 
of  the  thesis.  From  these  the  integrated  perturbations  were  obtained  by 
the  use  of  the  formulae  for  mechanical  integration  as  follows  : 

'/  —  ^    f1—  .1L  f'" 
•/"~-/0    ~y° 


720 

—  -         f  -U       - 
'          12  J*   r  240 


720 


^°'" 


The  elements  thus  derived  for  October  26.0,  the  date  chosen  to  trans- 
fer the  centre  of  motion  to  Jupiter,  are  as  follows: 

i=215°47'  0".6 
n=     2  37    7.1 
£1=   19    6  36.3 
i=     7  23  37.2 
log  a  =  0.5550265 
"    e  =  9.7209785 
"    ^  =  2.7174669 


>-  1890.0 


PART  SECOND 


TRANSFORMATION  OF  THE  CENTRE  OF  MOTION  FROM  THE  SUN  TO  JUPITER. 

La  Place,  in  the  fourth  volume  of  the  Mecanique  Celeste,  developes  a 
method  for  determining  the  perturbations  of  a  comet,  when  approaching 
very  near  a  planet.  This  method,  first  proposed  by  D'Alembert,  consists 
in  supposing  the  planet  to  have  a  sphere  of  activity,  within  which  the 
relative  motion  of  the  comet  is  affected  only  by  the  planet's  attraction 
and  beyond  which  the  absolute  motion  of  the  comet  about  the  sun  is  per- 
formed as  if  the  sun  alone  acted  upon  it.  The  radius  of  this  sphere 
depends  upon  the  mass  of  the  planet  and  its  distance  from  the  sun. 
This  method  is  very  simple  and  beautiful,  but  it  neglects  entirely  the 
effect  of  the  sun  as  a  disturbing  body  whilst  the  comet  is  traversing  its 
relative  orbit  about  the  planet. .  It  will  become  much  more  effective,  if 
we  merely  use  the  idea  of  the  sphere  of  activity  as  defining  approxi- 
mately the  point  at  which  we  may  conveniently  transpose  the  sun  and 
the  planet,  as  disturbing  force  and  central  force  respectively ;  and  after 
the  transformation  has  been  made,  we  may  treat  the  sun  and  the  comet 
as  bodies  revolving  around  the  planet  as  a  central  body ;  the  sun  acting 
as  a  disturbing  body  upon  the  comet.  The  perturbations  of  the  comet 
by  the  sun  may  be  computed  in  a  manner  entirely  similar  to  the  usual 
methods.  The  exact  point  in  the  comet's  orbit  at  which  the  transforma- 
tion is  made  is  of  no  great  importance,  provided  that  the  perturbations 
be  carefully  computed  both  before  and  after.  The  most  convenient  point 
will  be  that  given  by  La  Place's  idea  of  the  sphere  of  activity. 

To  obtain  the  radius  of  this  sphere  La  Place  proceeds  as  follows  :  Let 
r  and  r'  be  the  radii  vectores  of  comet  and  the  planet  respectively,  then 
the  sun's  action  on  the  comet  will  be  proportional  to  -^-,  and  that  of  the 
14 


POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889.  15 

/ 

planet  upon  the  comet  proportional  to  -T-, — ^.     When  the  comet  is  with- 
out the  sphere   of  activity   of  the  planet,  the   quantity  -j-  must  greatly 

/ 

exceed  ^ — r;.      Within   this   sphere  the  disturbing  action  of  the  sun  on 


the  comet,   which  is  proportional   to  —  ^  ---  ^  or  very   nearly  to    -^—^  —  - 


must   be   very  small  in  comparison  to  j—f  —  ^.      These  two  conditions  are 

satisfied  on  the  supposition  that  j—f  —  ^  is  a  mean  proportional  between  -j- 

2  ir'  _  r\ 
and    v       -  and  this  gives  for  the  radius  r'  —  r  =  p  of  the  sphere  of  activity 


2 
For  the  planet  Jupiter  I  found, 

log  -£-  =  8.73166 

And  that  day  on  which  the  values  of  p  and  r  most  nearly  satisfy  this 
equation  will  be  the  most  convenient  time  for  the  transformation.  This 
day  I  found  to  be  October  26.0,  1886,  at  which  time  we  have, 

log  p  =  9.45802 
"  r  =  0.72762 
«  L-  -  8.73040 

r 

Having  thus  determined  upon  the  time  of  change,  and  having  derived 
the  elements  about  the  sun  for  that  instant,  the  process  is  to  find  from 
these  elements  the  values  of  x,  y,  z,  D,x,  D,y,  and  D,z,  for  the  comet  referred 
to  the  sun.  Then  from  the  Ephemeris,  or  tables,  we  find  the  values  of 
the  corresponding  quantities  .r',  y',  z',  DJ',  Dy',  Dp',  for  Jupiter  referred  to 
the  sun.  Whence  for  the  relative  coordinates  of  the  comet  referred  to 
Jupiter  as  a  centre  we  have, 


1  =.  x  —  x'     D,XI  •=.  D,x  — 

i  =  y  —  y'   fyi  =  -%  — 

,  =  z  —  z'     D,ZI  =  D,z  — 


16  POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 

Then  from  these  quantities  we  can  determine  the  elements  of  the  relative 
orbit  about  Jupiter. 

FIRST.  To  find  the  values  of  the  relative  coordinates  and  their  dif- 
ferential coefficients. 

The  values  of  #,  y,  and  z  may  be  obtained  directly  from  the  elements 
by  the  usual  formulae,  and  the  values  of  the  differential  coefficients,  Dj, 
Dy,  and  D,z  by  the  corresponding  differential  formulae.  Assuming  the 
ecliptic  as  the  fundamental  plane,  with  the  positive  direction  of  the  axis 
of  x  directed  towards  the  vernal  equinox,  we  have,  following  the  notation 
used  by  Watson  in  his  Theoretical  Astronomy, 

x  =  r  sin  a  sin  (A  +  a) 
y  =.  r  sin  6  sin  (£  +  u) 
z  =  r  sin  t  sin  u 

Where  the  auxiliary  quantities,  a,  b,  A  and  B  are  the  constants  for  the 
ecliptic,  and  are  functions  of  H  and  *  alone.     They  are  determined  by  the 

formulae : 

cot  a  =  —  tan  ft  cos  i ;  cot  B  •=.  cos  fl  cos  f 

cos  ft  ,        siu  ft 

sin  a  =  - — T-  ;  sin  b  =  - — ^ 

sin  A  sin  J» 

Differentiating  the  above  expressions  for  x,  y  and  z,  with  regard   to   the 
time  we  have, 

=  —  Df  +  r  sin  a  cos  (A  +  u)  Z>,M 


r 


Dg  =  -?-  Df  +  r  sin  b  cos  (B  +  u)  D,u 
Df  •=  —  Df  +  r  sin  i  cos  u  Dtu 
The  values  of  Df  and  Dtu  may  be  found  from  the  equations  : 

r1  D,v  =  <?  cos  $  DtM 

Df  =  a  tan  <J>  sin  v  D,M 
where 


and 

Dtu  =  D,r 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889.  17 

Thus  from  the  elements  of  the  comet  for  October  26.0  as  before  given, 

I  find  for  that  date, 

o  =  192°  19'  5  .43 

«  =  175    49  36.23 
log     r  =  0.7276229 
"    2)^=6.9875183 
"    D/-  =  7.0784501* 

And  from  these  with  the  values  of  JT,  y  and  z  deduced  directly  from  the 

elements,  I  find, 

log      x  =  0.7126056  n 

"       y  =  0.1397898  * 

r  =  8.6991237 
"  1^=7.3945983 
=  7.6687772  « 
=  6.8307732* 


From  the  Britisk  Nautical  Almanac  for  1886,  after  reducing  the  lati- 
tude and  longitude  to  the  mean  equinox  and  equator  of  1890.0,  I  find  for 
the  position  of  Jupiter  on  October  26.0,  1886, 

/I'  =197°  39'  36'.  2 
P'=      1    17  45.6 
log  r'  =  0.7368541 

To  find  the  difierential  coefficients  of  these  quantities  I  took  from  the 
Almanac  seven  complete  positions  of  the  planet  separated  by  intervals  of 
twenty  days  and  reduced  each  to  the  mean  fixed  equinox  of  October  26.0 
by  applying  the  corrections  for  precession  and  nutation.  I  then  tabulated 
the  longitudes  and  found  the  differences  to  the  third  and  fourth  orders, 
and  from  these  by  the  formulae  of  interpolation  I  found  the  first  differ- 
ence, or  the  D&  for  October  26.0.  Similar  processes  gave  me  the 

and  Djr, 

=  -j-0°  4  31".81 

'  =  -0  0    0.95 
A  log  r'  =  —  0.000002545 

3 


18  POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889. 

Reducing  the  J)th  and  D$  to  radians,  I  have  for  the  differential  coefficients 
with  respect  to  the  respect  to  the  time, 

log  ^'  =  7.1198403 
"  Dtp'  =  4.6632985  n 
"    Dtr'  =  5.5047041  n 

To  find  #',  y',  z'  and  their  differential  coefficients,  we  have  the  ordinary 

formulae, 

x'  —  r'  cos  (3'  cos  X' 

y'  —  r'  cos  @'  sin  /I' 
z'  —  r'  sin  (3' 

and  the  corresponding  differential  formulae, 


x 


Dtx'  =  —  Dtr'  —  y'  Dfi  —  z'  cos 


—    y        7) «'     I     /y»'     7}  O  ' «/  CM  n    3 

__    — ~    ~LJ  n        T^  **/      J-/if\i  &      Olll    A 

r' 
^z'  =  sin  /^'  D,r'  +  r'  cos  /?'  Dtp' 

Substituting  the  values  already  given  I  find  for  the  rectangular  coordinates 
of  Jupiter  referred  to  the  sun,  and  their  differential  coefficients, 

log      a;' =  0.7157782  » 
y'  —  0.2187138  n 
"        z'  =  9.0912994 
'  =  7.3444712 
=  7.8350142  n 
"    Dp'  =  5.4123645  n 

From  these  with  the  values  already  given  for  the  rectangular  coor- 
dinates of  the  comet  referred  to  the  sun,  I  find  for  the  relative  coordinates 
of  the  comet,  and  their  differential  coefficients,  referred  to  Jupiter  as  a 

centre,  the  following, 

log      x,  -  8.5778238 

"       ^=9.4392746 
zt  =  8.8655648 
FX  =  6.4320641 
/1=  7.3374871 
^t= 6.8138767  « 


POOB,  The  Action  of  Jupiter  upon  Comet  F,  1889. 


19 


SECOND.  From  the  above  relative  coordinates  and  their  differential 
coefficients  to  find  the  relative  orbit  of  the  comet  about  Jupiter. 

The  following  integrals  result  from  the  equations  of  motion  of  one 
body  around  another  (Mec.  Cel.,  Livre  II) : 

_  xdy  —  ydx       xdz  —  zdx     ff ydz  —  zdy 

~~dt        '      ~~         dt       }C    ~        HOT 

ydydx       zdzdx 


P- 


-f  -  ^ 

-J       r 


at 


1  + 


xdxdz 


zdzdy 

~dP~ 

ydydz 

dP 


daf  +  dtf  +  dz* 


a 


where  c,  c7,  c",  f,  f',  f",  and  a  are  arbitrary  constant  quantities.  The  ele- 
ments of  the  orbit  are  the  arbitrary  constant  quantities  of  its  motion,  they 
are  consequently  functions  of  the  above  arbitrary  constant  quantities.  They 
may  be  derived  by  means  of  the  following  formulae : 


tan  a  —  —. 


tan    i  = 


cn  j-c 


where  1=  longitude   of  the  projection   of  the  perihelion   on   the  funda- 
mental plane. 

In  the  special  problem  of  finding  the  relative  orbit  of  the  comet 
about  Jupiter,  we  note  that  K1  now  becomes  the  acceleration  at  unit's 
distance  due  to  the  force  exerted  by  the  mass  of  Jupiter:  and  as  the 
forces  of  gravitation  are  directly  proportional  to  the  mass  of  the  attract- 
ing body,  this  becomes, 

k"  -  m'K2 


20  POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889. 

where  yfc2  is  the  acceleration  due  to  the  sun,  namely 

(8.23558144)2 
and  m!  is  the  ratio  of  the  mass  of  Jupiter  to  that  of  the  sun,  namely 

1 

1047.879 
whence 

log  3k'2  =  3.4508517 

Substituting  in  formulae  (P)  the  numerical  values  of  the  relative 
coordinates  and  of  their  differential  coefficients  as  before  derived  for  October 
26.0,  we  find, 

logc  =4.8989042  log  /=  2.2265767  n 

"   c'  =  4.6811876  n  "   /  =  3.4147952  n 

"   c"  =  5.2903906  n  "    a  -  8.9374383  n 

From  these  are  deduced, 

H  =  256°  11'  2".0 

i=    68    29    0.85 
1=266   17  26.37 
log  a  =  8.9374383  n 
"    e  =  0.0041062 
"    r  —  9.4580166 

and  the  relative  orbit  is  therefore  hyperbolic.     To  find  T,  the  time  of  peri- 
jo  vian  passage  we  have, 


cos 
whence  as  r,  a  and  e  are  now  known,  we  find 

cos*1  =9.3690526 

F=7Q°  28'  21".  Q 
Again, 

^  (t  _  T)  =  N=  eh  tan  F—  log  tan  (45°  +  4 

J 
where 

log  a,  =  9.6377843 
hence, 

~^v  =  7.9570527 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889.  21 

From  the  last  member  of  the  above  equation  is  found, 

N  =0.8968864 
whence, 

t—T—  98.95612  days 

and  consequently  the  peri-jovian  passage  took  place, 

1886,  July  19.04388  Gr.  M.  T. 
To  find  from  /  the  longitude  of  the  peri-jove  we  have,  putting 

/ —  £1  EE  co'  and  n  —  H  =E  o 

cot  o  rz  cot  o'  cos  i 
whence  as 

o'  =  10°  6'  24"  57 
we  have 

a  =  25°  55'  12".00 

Collecting  these  results  we  have  for  the  complete  hyperbolic  elements 
of  the  comet  about  Jupiter  on  October  26.0,  1886,  Gr.  M.  T.,  the  following : 

7i  =  282°  6'  14".0 

fl  =  256  11    2.0 

i=   68  29    0.8 

o  =   25  55  12.0 
log  a  —  8.9374383  n 
11    e  =  0.0041062 
"    v  =  7.9570527 

T=  1886,  July  19.04388  Gr.  M.  T. 

THIRD.     Solar  Perturbations. 

To  compute  the  perturbations  due  to  the  action  of  the  sun  on  the 
comet  whilst  the  latter  was  traversing  the  above  hyperbolic  orbit  about 
Jupiter,  I  made  use  of  the  ordinary  formula  for  the  perturbations  of  the 
rectangular  coordinates  as  given  by  Watson  in  the  eighth  chapter  of  his 
Theoretical  Astronomy.  The  equations  there  derived  for  the  second  dif- 
ferential coefficients  of  the  perturbations,  with  reference  only  to  the  first 
power  of  the  disturbing  force,  are  as  follows : 


22  POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 

f-  -    -p^  |3  ^-  £r  —  ( 

*o  L       'o 


>  Sr  -  fol 

f>  J 

and 


fk 

where,  m',  x',  y'  and  z'  are  respectively  the  mass  and  the   coordinates  of 
the  disturbing  body. 

In  the  problem  now  under  consideration  K1  has  the  value  already 
derived  for  the  Jovian  system,  and  m'  is  the  ratio  of  the  mass  of  the 
sun  as  disturbing  body,  to  that  of  Jupiter  as  central  body,  or  is  the 
reciprocal  of  the  mass  of  Jupiter  as  usually  given.  Whence  we  have, 

7  *>  79  1  /  7  9  7  9 

m'  •—  fa"  ™  vn  «**  n  f*Ti  f*o  wt   A*^  ~~  ff 

—  j  A-     —  //ev/l/jj  //t'  A-     —  /t/5 

F  (1  +  m)  =  mfil 
Substituting  numerical  values  these  become, 

log  m'P  =  6.47116 
log  F  (1  +  TO)  =  3.45085 

And  for  an  interval  o  the  above  equations  for  the  differential  co- 
efficients become  when  expressed  in  units  of  the  seventh  decimal  place: 


=  rf  [3.47116]  p^S  -  £]  +  »'  »  (3  S 
=  .'  [3.47116]  [t        -  ^  +  „'  »  (3  £ 


2   ro^^nftn         -°  2  [3.45085] 

r=«   [3.47116]    —-    - 


The  first  term  of  the  second  members  of  these  equations  can  be  com- 
puted directly  for  all  the  required  dates.  The  second  terms,  however, 
can  only  be  obtained  by  a  process  of  approximation  as  they  contain  the 
quantities  8r  and  8x  or  %  or  8z.  We  first  derive  approximate  values  of 


,   d?Sx      „    cP&y  ,    d?8z 

~dP'       ~d/  an          ~~dF     y  neglecting  the  second  terms.      These  are 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889.  23 

integrated  by  the  usual  formulae  for  integration  and  with  the  resulting 
approximate  values  of  &r,  ty  and  &z  we  complete  the  values  of  the  differ- 
ential coefficients  and  integrate  anew.  In  this  way  the  perturbations  may 
be  carried  back  from  date  to  date.  In  some  cases,  however,  two  and  even 
three  or  more  approximations  had  to  be  made. 

From  October  26th  to  August  17th,  a  ten  day  interval  was  used. 
But  on  the  latter  date  the  indirect  terms  in  the  above  equations  became 
too  large  for  convenience  or  for  accuracy.  It  was  necessary,  therefore,  to 
apply  the  perturbations  to  the  elements,  and  thus  to  obtain  a  new  set  of 
osculating  elements.  This  transformation  is  effected  by  means  of  the 
values  of  the  perturbations  of  the  coordinates,  with  the  corresponding 
values  of  the  variations  of  the  velocities, 

dx  dy  dz 

dT'  ~dt'  ~dt 

The  variations  of  these  quantities,  or 

<%  dSz 


d'  d'  d 

are  obtained  by  a   single   integration   from  the  second  differential  coeffi- 
cients already  obtained.     Then  we  have  for  the  date  required 

.   ,  dx        dxo   ,    dSx 

*=*„+&,       _  =  _  +  _ 

»=*+*       = 


dz         dzo        dSz 


0, 

From  these  values  the  new  osculating  elements  may  be  computed  by 
equations  (P)  as  before. 

To  find  the  values  of  —  °,  ~.  -  ?  from  the  hyperbolic  elements,  we  have 

'  *f          (  *(          (Z-C 

N  =  %N0         and'        ^0  =  e  tan  F  —  log£  tan  (45°  +  4-  F) 

A 

Differentiating  with  respect  to  the  time,  but  keeping  the  elements  constant, 
we  get, 


^_     _  —(__1  __  i\  dF     l 

dT~     ~dT'  ~dT  ~  Vcos  F         )   di  cos  F 


24  POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889. 

But  from  the  theory  of  hyperbolic  motion, 


s  F 
Substituting  in  the  above, 

dF       1  a 


~dt  '  sin  F  ~  r  tan  F'  ~dt 
As  the  elements  are  constant  we  have, 

dN 

—  —  v 
dt 


Taking  the  logarithms  of  both  members  of  the  equation, 
and  differentiating, 


tan  —  F=  tan  —  v  tan  — 

2t  8  •.  .2 


dv        sin  v      dF 


dt~    sin  F  '  dt 
And  from  the  above  value  of  r,  we  have  by  differentiating, 

dv  _  ae  tan2  F  dF 
~dt~     sin  F    '  ~dt 

Again,  u  =  v  +  n  —  £1,  whence 

du dv 

~dt~dt~ 

The  values  of  a,  y  and  z  and  D,x,  D,y  and  Dtz  may  now  be  obtained 
from  the  hyperbolic  elements  in  a  manner  entirely  similar  to  that  by 
which  the  corresponding  values  were  obtained  from  the  elliptic  elements 
on  October  26th.  That  is, 

x  rr  r  sin  a  sin  (A  +  w) 
y  —  r  sin  b  sin  (B  -\-  u) 
z  —  r  sin  c  sin  (C  +  u) 


and 


rt»  B 

=.  -  -  Dp  +  r  sin  a  cos  (A  +  w) 
D,y  —  —  J)tr  +  r  sin  b  cos  (J5  +  u)  Dtu 
Dtz  zr  -  -  i>,r  -f-  r  sin  i  cos  i 


POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889.  25 

From  the  osculating  hyperbolic  elements  of  October  26th,  we  thus  find 
for  August  17th  : 

F=    64°   OM6  log    r-  9.05169 

v  =  167   26.54  "  D,u  =  7.23045 

u  =  193   21.74  "  Dtr  =  7.45804 

and 

log  OTO  =  8.22788  log  DPrQ  =  6.54880 

"   y0  =  9.03623  "   £#»  =  7.43838 

"    z0  =  8.38413  n  "    ^0  =  6.89795  n 


From  the  table  of  perturbations,  computed  as  before  explained,  we  have 
for  August  17th  by  integrating: 

log  8x  =  6.99648  log  D£x  =  5.35031  n 

"    %  =  6.86291  n  "    D$y  —  5.29714 

"    8z  =  6.39262  "    V£z  =  4.79810  n 


Applying  these  to  the  values  of  #0,  y0,  zc,  etc.,  we  have  for  the  corrected 
values  of  the  rectangular  coordinates  and  their  differential  coefficients  for 
August  17.0  : 

log  x  =  8.25265  log  Dj  =  6.52040 

"    y  =  9.03331  "    Dty—  7.44151 

"    z  —  8.37968  n  "    Dp  =  6.90139  n 

From  these  were  deduced  the  new  elements,  as  follows  : 

n  —  281°  41'.49 

11  =  252    18.49 

i=    56    26.41 

u=    29    23.00 
log  a  —  8.92681  n 
"    e  —  0.00549 
"  v  —  7.97300 

T-  July  19.3273  Gr.  M.  T. 

With  these  as  osculating  elements  the  perturbations   were   continued 
until  July  20th,  using  a  four  day  interval.      As  the  peri-jove  is  approached 
4 


26 


POOR,  The  Action  of  JupHer  upon  Comet  V,  1889. 


rl  becomes  a  very  small  quantity,  and  hence  the  coefficient  of  the  indirect 
term  of  the  differential  coefficient  becomes  very  large  ;  and  the  term  itself 
more  and  more  difficult  to  approximate  to  in  value.  It  was  necessary, 
therefore,  to  again  apply  the  perturbations  and  to  deduce  new  elements 
for  July  24th.  The  perturbations  were  then  continued  until  July  4th, 
when  they  were  again  applied  and  new  osculating  elements  obtained. 
With  these  elements  of  July  4th,  the  perturbations  were  carried  back, 
using  a  period  of  ten  days,  to  March  26th,  on  which  day  the  comet 
passed  out  of  the  sphere  of  Jupiter's  activity.  These  various  sets  of  ele- 
ments are  given  below,  and  the  perturbations  are  given  in  the  tables  at 
the  end  of  the  thesis  : 

HYPERBOLIC  ELEMENTS. 

Oct.  26.0 

6'  14".3 
11    20.0 

i—  68  29  0.8 
o=  25  55  12.3 
log  a  —  8.9374383  n 
"  e-  0.0041062 
"  v-  7.9570528 
T—  Julv  19.0439 

/ 


Aug.  17.0 
281°  41'.49 

July  24.0 
281°  44'.50 

252  18.49 

252  11.36 

56  26.41 

56  1.63 

29  23.00 

29  33.14 

8.92681  n 

"  8.92450  n 

0.00549 

0.00568 

7.97300 

7.97646 

19.3273 

19.3306 

I  — 


o  = 
log  a  — 

"    e  — 


July  4.0 
=  281°  42'.85 
=  252   12.28 
56     2.00 
29   30.57 
8.92423  n 
0.00558 
7.97686 


V  — 


T—  19.3296 


March  26.0 
283°  48'.33 
255     8.06 
58   55.36 
28    40.27 
8.94800  n 
0.00479 
7.94121 
20.1238 


From  these  we  see  that  the  solar  perturbations  produce  quite  marked 
changes  in  the  relative  hyperbolic  orbit  about  Jupiter.     For  nearly  a  month 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889.  27 

of  the  closest  approach,  however,  these  perturbations  are  very  small,  and 
but  very  slightly  affect  the  character  of  the  resulting  orbit.  During  this 
time  also  the  computation  is  very  difficult,  owing  to  the  very  rapid  motion 
of  the  comet  in  its  orbit,  and  to  its  very  small  distance  from  the  planet. 
A  very  good  approximation  to  the  effect  of  the  solar  perturbations  can  be 
obtained  by  neglecting  entirely  the  action  of  the  sun  as  a  disturbing  body 
for  perhaps  two  weeks  before  and  after  the  comet  passed  its  peri-jove. 

The  remarkable  character  of  the  appulse  is  seen  by  the  elements  of 
July  24.0.  The  comet  passed  the  centre  of  Jupiter  on  1886,  July  19.33, 
at  no  greater  distance  than  two  and  a  third  radii  of  that  planet.  It  must 
then  have  passed  the  surface  of  the  planet  at  a  distance  of  only  one  and 
a  third  radii ;  that  is  the  centre  of  the  comet  was  only  57,650  miles  from 
the  surface  of  the  planet.  It  is  not  at  all  improbable  that  parts  of  the 
diffused  mass  of  the  comet  may  have  swept  over  the  surface  of  the  planet 
itself;  and  this,  together  with  the  unequal  attractions  of  the  planet's  sat- 
ellites upon  the  different  parts  of  the  comet  may  have  tended  to  disrup- 
tion, and  caused  the  separation  actually  observed. 

Two  diagrams  ar-e  given,  showing  the  character  of  the  orbit  about 
Jupiter.  The  first  of  these,  A,  represents  the  projection  of  the  relative 
hyperbolic  orbit  about  Jupiter  upon  the  plane  of  the  ecliptic.  The  figure 
also  contains  the  orbits  of  the  two  outer  satellites  of  Jupiter.  Figure  B 
shows  the  true  paths  of  Jupiter  and  the  comet  projected  upon  the  plane 
of  the  ecliptic.  The  slightly  curved  line  represents  that  portion  of  Jupi- 
ter's orbit  traversed  between  March  and  October  1886;  on  this  the  posi- 
tions of  the  planet  for  various  intermediate  dates  are  shown.  The  full 
curved  line  represents  the  actual  orbit  of  the  comet  during  the  same 
period,  and  on  it  are  shown  the  various  positions  of  the  comet  for  the 
same  dates.  On  March  26th  the  comet  is  shown  just  entering  the  sphere 
of  Jupiter's  activity,  at  a  point  outside  the  orbit  of  that  planet;  as  the 
two  bodies  proceed  the  comet  approaches  closer  and  closer  to  Jupiter 
until  July  20th,  when  it  attains  its  nearest  approach.  At  this  time  it 
crosses  the  planet's  orbit  and  henceforth  continues  its  course  within  the 
orbit  of  Jupiter,  and  is  shown  leaving  the  sphere  of  activity  on  October 
26th.  Near  the  point  of  closest  approach  is  found  a  point  of  inflection 
in  the  comet's  orbit. 


PART  THIRD. 


TRANSFORMATION    FROM   JUPITER    TO    THE    SUN    AS    CENTRAL    BODY. 

The   date   chosen   for   this   transformation  was  March  26.0.  at  which 
time  the  comet  was  distant  from  Jupiter, 

log  p  =  9.5128642 

The  process  of  the  transformation  is  entirely  similar  to  that  already  ex- 
plained when  we  passed  from  the  sun  to  Jupiter  on  October  26.0.  From 
the  hyperbolic  elements  about  Jupiter  are  derived  the  rectangular  coor- 
dinates and  their  differential  coefficients  referred  to  Jupiter  as  a  centre. 
From  these  and  from  the  corresponding  quantities  for  Jupiter  referred  to 
the  sun  as  centre  are  obtained  the  coordinates  of  the  comet  and  their 
differential  coefficients  referred  to  the  sun.  The  elements  of  the  comet's 
orbit  about  the  sun  are  then  derived  by  means  of  equations  P. 

For  March  26.0  the  relative  coordinates  of  the  comet  and  their  dif- 
ferential coefficients  referred  to  Jupiter  as  a  centre  are: 

log  x  -  8.5792316  n  log  Dtx  =  6.3416455 

"    y  —  9.4310521  "    Dty-  7.2748372  n 

"     *  =  9.2448044  n  "     Dtz  =  7.0621004 

From  the  British  Nautical   Almanac  we  find,  in  the   manner  already 
explained  (page  17),  for  the  position  of  Jupiter  on  March  26.0^ 

Jl'=      181°  30'  10".  1 
p'=          1    17   51.1 
log  r'  =      0.7365048 

+     0°    4'  32".3 
+     0     0     0.85 
log  Dtr=      5.8621048 
28 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889.  29 


Reducing  Dfa  and  D$  to  radians,  and  substituting  as  before,  we  find, 
for  the  rectangular  coordinates  of  Jupiter  referred  to  the  sun  and  their 
differential  coefficients, 

log      x1  =  0.7362440  n 

if  -  9.1551237  » 

z'  =  9.0914617 
"     DX  =  6.0661227 
"     D,y'  =  7.8580167  n 
"     Dtz'  =  5.3821473 

Combining  these  with  the  above  given  relative  coordinates  of  the 
comet  referred  to  Jupiter,  we  find,  for  the  rectangular  coordinates  and  their 
differential  coefficients  of  the  comet  referred  to  the  sun  as  centre, 

log    x  =  0.7392581  n 
y  —  9.1033802 
2  =  8.7182655  n 
"  ^  =  6.5264067 
"  Dty  =  7.9587691  n 
"   7^  =  7.0710816 

Substituting  these  values  in  formulae  (P)  and  remembering  that  Xtr 
now  has  its  usual  value,  i.  e.,  the  acceleration  at  unit's  distance  due  to 
the  force  exerted  by  the  sun,  we  have, 

log  c  —  8.6976559  log  /  =  6.1977764  n 

"    <f  =  7.8091574  n  "   /'  =  5.3797986  n 

"    c"  =  6.5131308  n  "     a  =  1.0979460 

From  these  are  deduced  : 

H  =  182°  53'  43".86 
i=     7  22   30.55 
7=188  38   46.7 
log  a  —  1.0979460 
"    e  =  9.75139  10 
"    r  =  0.7393938 
"       =  1.9030876 


30 


POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 


The  resulting  orbit  is  therefore  elliptic.      To   find  the   time  of  peri- 
helion passage  we  have  from  the  theory  of  elliptic  motion, 


COS  J£=. 


and 

Whence  we  have, 


a —  r 

ae 


M=E—e"  sin  E 

E=  —  4?  58'  2"  .5 
M=-2  10   6.8 
t—  T-  —  97.5855  days. 

Finding  n  from  I  as  before,  we  have  for  the  elliptic  elements  of  the 
comet  about  the  sun  on  March  26.0,  the  following: 


n  =  188°  41'  38".2 
H  =  182   53   43.8 
i=     7    22  30.6 
u  =     5   47   54.4 
log  a  =  1.0979460 

"     e  —  9.7513910 

"    ^  =  1.9030876 

T—  1886  July  1.5855 


1890.0 


PART  FOURTH. 


While  the  results  here  derived  confirm  Mr.  Chandler's  conclusions  as 
to  the  identity  of  the  two  comets,  yet  they  differ  radically  from  his  in 
several  very  important  particulars.  The  comet  is  found  to  have  approached 
Jupiter  much  closer  than  he  suspected,  and  the  resulting  changes  in  the 
orbit  are  much  more  radical.  Before  the  appulse  in  1886,  the  orbit  is 
found  to  be  much  larger  than  his  work  seemed  to  indicate,  the  period 
being  over  forty  years  instead  of  only  twenty-seven.  The  position  of  the 
orbit  in  space,  however,  is  but  slightly  different  from  that  deduced  by 
him;  the  inclination  and  the  line  of  nodes  are  practically  unaltered,  the 
longitude  of  the  perihelion  is  changed  but  a  few  degrees  from  the  value 
assigned  in  his  paper. 

The  radical  character  of  the  changes  in  the  comet's  orbit,  caused  by 
the  appulse,  are  shown  in  Plate  II.  Here  are  represented  the  orbits  of 
the  earth,  Jupiter,  and  Saturn,  together  with  those  of  the  comet  before 
and  after  its  approach  to  Jupiter  in  1886.  The  present  orbit  of  the  comet 
is  the  small  full  ellipse,  with  its  perihelion  beyond  the  earth's  orbit,  and 
its  aphelion  just  outside  the  orbit  of  Jupiter  in  longitude  180°  approxi- 
mately. On  this  curve  the  positions  of  the  comet  are  shown  at  the  time 
of  discovery  on  July  6th,  1889,  at  perihelion  passage,  and  at  the  date  of 
the  last  observation  in  December,  1890.  The  large  heavy  ellipse  reaching 
far  beyond  the  orbit  of  Saturn  is  the  orbit  of  the  comet  before  its  dis- 
turbance in  1886.  These  two  paths  are  tangent  to  each  other  near  the 
point  of  closest  approach  in  heliocentric  longitude  180°.  The  disturbance 
completely  reversed  the  orbit,  the  perihelion  before  that  date  coincides 
almost  exactly  with  the  present  aphelion.  The  large  old  orbit  was  one 
that  the  comet  traversed  in  about  forty  years;  the  present,  one  of  only 
seven  years.  On  this  large  curve  is  shown  the  position  of  the  comet  in 
March,  1886,  as  it  was  approaching  conjunction  with  Jupiter.  This  orbit 

31 


32  POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 

shows  that  the  comet  passed  very  close  to  the  orbit  of  Saturn  in  helio- 
centric longitude  90°  and  290°,  and  that  at  either  one  of  these  points  it 
is  possible  to  have  a  close  approach  of  the  two  bodies.  The  plate  also 
shows  the  original  orbit  of  Lexell's  comet  as  a  small  dotted  ellipse  of 
large  eccentricity  with  its  aphelion  coinciding  very  nearly  with  the  aphe- 
lion of  the  present  orbit  of  Comet  V. 

The  point  upon  which  Mr.  Chandler  laid  so  much  stress,  that  the 
period  of  the  comet  before  1886  was  approximately  26.5  years,  seems  to 
be  overthrown.  Assuming  the  substantial  correctness  of  his  period  he 
showed  that  four  periods  of  the  comet  (107.8  yrs.)  are  nearly  equal  to 
nine  of  Jupiter's  (106.9  yrs.),  and  that  the  two  bodies  would  have  had, 
therefore,  another  close  approach  in  1779.  His  conclusions  as  to  the 
identity  of  Comet  V  1889  and  Lexell  1770,  depend  entirely  upon  the 
assumption  of  four  revolutions  of  the  comet  during  the  hundred  and  seven 
years  between  the  appulse.  For,  according  to  him,  three  revolutions  dur- 
ing this  period  would  make  one  period  of  the  comet  very  nearly  equal  to 
three  of  Jupiter's  (35.6  yrs),  and  cause  close  approaches  of  the  two  bodies 
in  1850  and  again  in  1875.  Such  approaches  and  the  consequent  enor- 
mous changes  in  the  orbit  render  all  attempts  to  determine  the  character 
of  the  orbit  in  1779  utterly  fruitless ;  and  consequently  with  the  suppo- 
sition of  thi'ee  revolutions  of  equal  periods  between  1886  and  1779,  we 
cannot  hope  to  establish  the  identity  of  these  two  remarkable  bodies. 

My  work,  which  gives  a  period  of  about  forty-four  (44)  years  for  the 
comet  before  1886,  seems  to  indicate  only  three  revolutions  in  the  107 
years  to  be  accounted  for  in  order  to  establish  identity,  but  revolutions 
of  unequal  periods  owing  to  large  perturbations  by  Saturn.  The  comet, 
according  to  my  results,  was  at  its  shortest  distance  from  Saturn's  orbit 
about  1846,  in  heliocentric  longitude  295°,  and  Saturn  was  at  the  same 
point  in  its  orbit  about  1844.7.  While  I  have  not  had  time  to  determine 
accurately  the  character  of  this  approach,  yet  it  seems  probable  that  the 
resulting  changes  in  the  comet's  orbit  were  large  and  sufficient  to  have 
changed  the  period  by  a  few  years.  I  made  a  very  hurried  and  rough 
approximation  to  the  effect  of  the  perturbations  of  Jupiter  for  a  few 
months  before  the  appulse  in  1886,  and  also  as  to  the  character  of  the 


POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889.  33 

perturbations  by  Saturn  in  1846,  and  found  that  the  period  was  shortened 
considerably.  This  would  seem  to  indicate  a  little  more  than  two  com- 
plete revolutions  of  about  34  years  each  between  1779  and  1846,  and  a 
nearly  complete  revolution  of  about  44  years  from  that  date  to  1886. 
Mr.  Schulhof  arrived  at  practically  the  same  conclusions  by  an  entirely 
different  method. 

His  first  paper  appeared  in  the  Bulletin  Astronomique,  November, 
1889.  He  there  discussed  the  possibility  of  the  identity  of  several  pairs 
of  periodic  comets,  by  means  of  a  criterion  formulated  by  M.  Tisserand. 
A  quantity,  n,  is  derived,  which,  if  two  comets  seen  at  different  appari- 
tions are  identical,  must  be  the  same.  The  value  of  n  is  given  by  the 

formula, 

.  1       2VA 
n  —  ~~r  -Jp-  */p  COS  I 

where,  a,  p  and  i  are  respectively  the  semi-major  axis,  parameter  and 
inclination  of  the  orbit,  and  A  and  E  the  semi-major  axis  and  radius 
vector  of  any  disturbing  planet  at  the  point  of  nearest  approach.  For  a 
single  comet  the  perturbations  by  a  single  planet  can  produce  only  a 
small  variation  in  the  value  of  ».  Mr.  Schulhof  finds  (Bulletin  Astro- 
nomique, December,  1889),  using  Le  Terrier's  elements  of  Lexell's  comet 
and  Chandler's  of  Comet  V,  that  this  criterion  can  only  be  satisfied  for 
these  two  comets  upon  the  supposition  of  a  strong  perturbation  by  Saturn. 
Assuming  the  identity  of  the  two  comets,  he  deduces  by  means  of  the 
criterion  the  most  probable  orbit  of  the  comet  between  1779  and  1886,  and 
finds  its  period  to  have  been  about  32  years  from  17J9  to  1849,  at  which 
time  the  perturbations  of  Saturn  increased  its  period  to  about  42  years. 
This  agrees  so  strikingly  with  the  results  of  my  direct  computation 
of  these  intermediate  orbits,  that  there  can  be,  I  think,  no  doubt  as  to 
the  identitv  of  these  two  comets. 


CHARLES  LANE  POOR,  the  third  son  of  Edward  Eri  and  Mary  (Lane) 
Poor,  was  born  at  Hackensack,  New  Jersey,  on  January  18th,  1866. 
After  studying  in  private  and  public  schools  he  entered  the  Introductory 
Department  of  the  College  of  the  City  of  New  York  in  the  fall  of  1880 
and  was  graduated  from  that  Institution  with  the  degree  of  Bachelor  of 
Sciences  in  June,  1886.  The  following  October  he  was  appointed  a  Tutor 
in  that  college.  This  position  he  held  for  two  years,  teaching  Descriptive 
Geometry  and  Mathematics. 

In  October,  1888,  he  resigned  the  above  position  to  enter  the  Johns 
Hopkins  University,  where  he  became  a  student  of  Astronomy  and  Physics. 
In  January,  1889,  he  was  appointed  University  Scholar  in  Astronomy, 
and,  at  the  following  Commencement,  Fellow  in  Astronomy. 

The  degree  of  Master  of  Sciences  was  conferred  on  him  by  the  Col- 
lege of  the  City  of  New  York  in  June,  1889. 

During  his  graduate  study  he  has  attended  courses  given  by  Pro- 
fessors Simon  Newcomb,  Henry  A.  Rowland  and  Doctors  Craig  and  Kim- 
ball,  all  of  the  Johns  Hopkins  University. 


34 


TABLES. 


PERTURBATIONS   BY  JUPITER. 

d<f> 

t_f                                    cHj 
J                             <*  -£ 

'/ 

1886,  Oct.  26, 

-  1306.00 

+  9376.40 

-  892.70 

+  7157.20 

Nov.    5, 

-  1116.80 

+  8070.40 

-  783.20 

+  6264.50 

15, 

964.60 

+  6953.60 

-  692.70 

+  5481.30 

25, 

-   858.60 

+  5989.00 

—  620.30 

+  4788.60 

Dec.     5, 

-   758.50 

+  5130.40 

-  562.20 

+  4168.30 

+  4371.90 

+  3606.10 

15, 

-    672.80 

-  512.10 

25, 

591.80 

+  3699.10 

-457.80 

+  3094.00 

1887,  Jan.     4, 

545.60 

+  3107.30 

-  432.70 

+  2636.20 

+  2561.70 

+  2203.50 

14, 

497.00 

-  402.10 

+  2064.70 

+  1801.40 

24, 

452.90 

-  377.10 

+  1611.80 

+  1424.30 

Feb.    3, 

-   414.50 

-351.40 

+  1197.30 

+  1072.90 

13, 

-   380.00 

-330.20 

+   817.30 

+    742.70 

23, 

348.60 

-  310.30 

+   468.70 

+   432.40 

Mch.  5, 

-   319.30 

-292.90 

+    149.40 

+    139.50 

15, 

294.90 

+      (2.00) 

-  276.50 

+      (1-25) 

25, 

-   270.80 

—  261.90 

35 


36 


1887,  Mch.  15, 
Apr.  24, 
June    3, 
July  13, 
Aug.  22, 
Oct.     1, 
Nov.  10, 
Dec.  20, 

1888,  Jan.  29, 
Mch.   9, 
Apr.  18, 
May  28, 
July    7, 
Aug.  16, 
Sept.  25, 
Nov.    4, 
Dec.  14, 

1889,  Jan.   23, 
Mch.    4, 
Apr.  13, 
May  23, 
July     2, 
Aug.  11, 
Sept.  20, 


OOR,  The  Action 

of  Jupiter  upon  Comet  V,  1889. 

d<f> 

,f           dL 

"  "eft 

f          "Tt 

'f 

+  4942.59 

+  6302.13 

-  1032.82 

—  958.26 

+  3909.77 

+  5343.87 

t,    -  795.59 

-  811.32 

+  3114.18 

+  4532.45 

t,    -  622.67 

-  698.62 

+  2491.51 

+  3833.83 

^    _  494.96 

-  610.55 

+  1996.55 

+  3223.28 

t,      394.42 

-  533.46 

+  1602.13 

+  2689.82 

-  317.39 

-  472.09 

7 

+  1284.74 

+  2217.73 

>,    —  254.26 

-  412.43 

+  1030.48 

+  1805.30 

),      204.29 

-  360.06 

+  826.19 

+  1445.24 

>,    -  164.14 

-  311.22 

+  662.05 

+  1134.02 

>,      131.39 

-  265.05 

+  530.66 

+  868.97 

*,   —  105.26 

-  221.79 

+  425.40 

+  647.18 

t   -  84.68 

-  181.68 

7 

+  340.72 

+  465.50 

-  68.58 

-  144.82 

+  272.14 

+  320.68 

J,   -  56.21 

-111.89 

7 

+  215.93 

+  208.79 

5,       46.75 

-  83.10 

+  169.18 

+  125.69 

t,    -  39.56 

-  58.81 

7 

+  129.62 

+  66.88 

t,   -  35.38 

-  39.70 

7 

+  94.24 

+  27.18 

3,       28.88 

-  23.61 

+  65.36 

+  13.57 

1,      24.43 

-  12.12 

+  40.93 

+   1.45 

3,    —  20.00 

4.12 

+  20.93 

2.67 

3,   -  15.33 

+  0.97 

~^7 

+   5.60 

-   1.70 

2,       10.36 

+   (0.42) 

+  3.73 

+   (0.16) 

1,       5.33 

+  4.79 

0,       0.69 

+  4.72 

POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 

37 

^ 

'/ 

"f            6>  — 

| 

+  180.68 

+  3470.25 

1886,  Oct.  26, 

—  25.50 

-  879.92 

-  483.10 

+  155.18 

+  2987.15 

Nov.  5, 

-  21.71 

—  724.74 

-  417.30 

+  133.47 

+  2569.85 

15, 

-  18.67 

-  591.27 

-  363.65 

+  114.80 

+  2206.20 

25, 

-  16.75 

-  476.47 

-  315.10 

+  98.05 

+  1891.10 

Dec.  5, 

-  14.72 

-  378.42 

-  280.70 

+  83.33 

+  1610.40 

15, 

-13.00 

-  295.09 

—  239.80 

+  70.33 

+  1370.60 

25, 

-  11.46 

—  224.76 

-  217.30 

+  58.87 

+  1153.30 

1887,  Jan.  4, 

-  10.51 

-  165.89 

-  201.80 

+  48.36 

+  951.50 

H 

-  9.55 

-  117.53 

-  182.50 

+  38.81 

+  769.00 

24, 

-  8.65 

-  78.72 

-  167.50 

+  30.16 

+  601.50 

Feb.  3, 

7.84 

—  48.56 

-  154.00 

+  22.32 

+  447.50 

13, 

7.10 

—  26.24 

-  141.80 

+  15.22 

+  305.70 

23, 

6.51 

-  11.02 

-  129.90 

+  8.71 

+  175.80 

Mch.  5, 

—  5.95 

-  2.31 

-  119.50 

+  2.76 

+  56.30 

•   15, 

-  5.45 

+  (0.04) 

+  (0.45) 

-  111.30 

+  (0.70) 

25, 

5.01 

-  102.30 

+  624.51 

+  178.06 

+  1731.89 

1887,  Mch.  15, 

-  78.49 

+  802.57 

-  363.58 

+  99.57 

+  1368.31 

Apr.  24, 

-  58.33 

+  902.14 

-  286.85 

+  41.24 

+  1081.46 

June  3, 

-43.42 

+  943.38 

—  229.27 

-  2.18 

+  852.19 

July  13, 

-  33.26 

+  941.20 

—  185.65 

—  35.44 

+  666.54 

38 


POOR, 


The  Action  of  Jupiter  upon  Comet  V,  1889. 

di 


'  dt 

j 

fiT 

j 

1887,  Aug.  22, 

-23.12 

+  905.76 

-  150.07 

-58.56 

+  516.47 

Oct.   1, 

-  15.77 

+  847.20 

-  122.03 

-  74.33 

+  394.44 

Nov.  10, 

-  9.87 

+  772.87 

-  97.79 

-84.20 

+  296.65 

Dec.  20, 

4.78 

+  688.67 

78.17 

-  88.98 

+  218.48 

1888,  Jan.  29, 

-  0.69 

+  599.69 

-  61.68 

-89.67 

+  156.80 

Mch.  9, 

+  2.72 

+  510.02 

47.65 

-86.95 

+  109.15 

Apr.  18, 

+  5.41 

+  423.07 

35.97 

-81.54 

+  73.18 

May  28, 

+  7.42 

+  341.53 

26.41 

-  74.12 

+  46.77 

July  7, 

+  8.81 

+  267.41 

18.68 

-65.31 

+  28.09 

Aug.  16, 

+  9.62 

+  202.10 

12.63 

-55.69 

+  15.46 

Sept.  25, 

+  9.88 

+  146.41 

-  8.05 

-45.81 

+  7.41 

Nov.  4, 

+  9.69 

+  100.60 

4.72 

—  36.12 

+  2.69 

Dec.  14, 

+  9.19 

+  644.48 

2.49 

-26.93 

+  0.20 

1889,  Jan.  23, 

+  8.21 

+  37.55 

-  0.97 

-  18.72 

0.77 

Mch.  4, 

+  7.08 

+  18.83 

0.11 

-11.64 

-  0.88 

Apr.  13, 

+  5.76 

+  7.19 

+  0.29 

-  5.88 

—  0.59 

May  23, 

+  4.33 

+  1.31 

+  0.41 

-  1.55 

—  0.18 

;  July  2, 

+  2.87 

-  (0.12) 

-  (0.24) 

+  0.35 

-  (0.01) 

Aug.  11, 

+  1.44 

+  0.20 

Sept.  20, 

+  0.19 

+  0.03 

POOR,  The  Action  of  Jupiter  upon 
dfl 


Comet  F,  1889. 

• 

dir 


39 


-  719.60 

+  2445.70 

1886,  Oct. 

26, 

+  280.10 

-230.40 

-  439.50 

+  2215.30 

Nov 

.  5, 

+  209.60 

-  214.20 

—  229.90 

+  2001.10 

15, 

+  154.40 

-  205.40 

-  75.50 

+  1795.70 

25, 

+  121.70 

-  195.10 

+  46.20 

+  1600.60 

Dec. 

5, 

+  84.90 

-  188.40 

+  131.10 

+  1412.20 

15, 

+  52.40 

-  176.80 

+  183.50 

+  1235.40 

25, 

+  31.20 

-  168.70 

+  214.70 

+  1066.70 

1887,  Jan. 

4, 

+  11.20 

-  163.70 

+  225.90 

+  903.00 

14, 

-  3.20 

-  159.70 

+  222.70 

+  743.30 

24, 

16.80 

-  151.60 

+  205.90 

+  591.70 

Feb. 

3, 

28.40 

-144.00 

+  177.50 

+  447.70 

13, 

39.40 

-  136.30 

+  138.10 

+  311.40 

23, 

49.70 

-  130.20 

+  88.40 

+  18120 

Mch. 

5, 

-  56.00 

-  122.70 

.+  32.40 

+  58.50 

15, 

-  66.00 

-  (0.65) 

-  116.20 

+   (0.40) 

25, 

-  71.90 

-  111.20 

+  4737.22 

+  1579.45 

1887,  Mch. 

15, 

-  292.76 

-  416.80 

+  4444.46 

+  1162.65 

Apr. 

24, 

-  356.60 

-  363.68 

+  4087.86 

+  798.97 

June 

3, 

-  388.12 

-  311.95 

+  3699.74 

+  487.02 

July 

13, 

-  402.92 

-  262.64 

+  3296.82 

+  223.38 

Aug. 

22, 

-  401.88 

-  217.50 

+  2894.94 

+   5.88 

40 


1887,  Oct.     1, 
Nov.  10, 
Dec.  20, 

1888,  Jan.  29, 
Mch.   9, 
Apr.  18, 
May  28, 
July    7, 
Aug.  16, 
Sept.  25, 
Nov.    4, 
Dec.  14, 

1889,  Jan.  23, 
Mch.    4, 
Apr.  13, 
May  23, 

'  July  2, 
Aug.  11, 
Sept.  20, 


3R,  The  Action  of  Jupiter  upon 
dfi       ,/. 

-  393.40 
+  2501.54 

»,   -  373.27 

+  2128.27 
,    -  349.56 
+  1778.71 
,  •  -  320.00 
+  1458.71 
,    -  287.81 

Comet  F,  1889. 

0  #" 
-  171.96 

-  130.56 
-  91.43 
—  56.31 
-  23.97 

-  166.08 
-  296.64 
-  388.07 
-444.38 

+  1170.90 

-  468.35 

,   —  252.80 

+ 

3.63 

+  918.10 

—  464.72 

«,    —  216.85 

+ 

26.01 

+  701.25 

-  438.71 

-  181.00 

+ 

42.99 

+  520.25 

—  395.72 

>,    -  146.94 

+ 

54.39 

+  373.31 

-  341.33 

-  115.54 

+ 

61.19 

+  257.77 

-  280.14 

-  87.75 

+ 

61.19 

+  170.02 

-  218.95 

-  65.47 

+ 

58.84 

+  104.55 

-  160.11 

},   _  44.70 

+ 

51.54 

+  59.85 

-  108.57 

[,   -  29.44 

+ 

42.96 

+  30.41 

65.61 

I,   -  17.96 

+ 

33.33 

+  12.45 

—  32.28 

},   -  9.84 

+ 

23.84 

+   2.61 

—  8.44 

5,       4.53 

+   (0.35) 

+ 

15.72 

-  (0.58) 

L,   -  1.47 

+ 

9.91 

),   -  0.11 

+ 

7.00 

POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889.  41 


SOLAR  PERTURBATIONS. 

<PBx 


+  359.89 

Mch.  26, 

-  126.72 

-  332.27 

-342.83 

+  233.17 

Apr.     5, 

97.20 

99.10 

-  107.20 

+  135.97 

15, 

-   71.04 

+   36.87 

+   30.95 

+   64.93 

25, 

48.84 

+  101.80 

-f   97.73 

+    16.09 

May     5, 

-    30.36 

+  117.89 

+  115.36 

14.27 

15, 

14.76 

+  103.62 

+  102.39 

-   29.03 

25, 

2.64 

+    74.59 

+    74.37 

31.67 

June    4, 

+     6.12 

+   42.92 

+   43.43 

-   25.55 

14, 

+    11.22 

+    17.37 

+    18.30 

- 

14.30 

24, 

+   10.68 

+     3.07 

+     3.96 

-     3.62 

July    4, 

+     6.56 

-    (0.34) 

—    (0.55) 

+     0.00 

+    15.94 

July     4, 

+     1.38 

-   66.63 

66.51 

+    17.32 

8, 

+     0.72 

49.31 

49.25 

+    18.04 

12, 

-     0.02 

31.27 

31.272 

+    18.02 

16, 

2.40 

13.25 

13.45 

+   15.62 

20, 

18.72 

+     2.37 

+     0.81 

-     3.10 

24, 

+     8.79 

+    (1-30) 

(0.73) 

0.00 

+     5.70 

28, 

+   12.60 

+     4.97 

6 


42  POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 


df 

./ 

/ 

\JtA> 

2.07 

July  24, 

-  74.04 

+  287.18 

+  281.01 

-  76.11 

28, 

-  0.72 

+  211.07 

+  211.01 

-  76.83 

Aug.  1, 

+  12.84 

+  134.24 

+  135.31 

-  63.99 

5, 

+  17.06 

+  70.25 

+  71.67 

-  46.93 

9, 

+  21.28 

+  23.32 

+  24.09 

—  25.65 

13, 

+  25.50 

2.33 

0.21 

-  0.15 

17, 

+  29.72 

2.48 

0.00 

-2191.60 

Aug.  17, 

—  64.20 

+  9924.60 

+  9919.25 

-  2255.80 

27, 

+  136.44 

+  7668.80 

+  7680.17 

-  2119.36 

Sept.  6, 

+  263.16 

+  5549.44 

+  5571.37 

-  1856.20 

16, 

+  314.52 

+  3693.24 

+  3719.45 

-1541.68 

26, 

+  377.28 

+  2151.56 

+  2183.00 

-  1164.40 

Oct.  6, 

+  401.64 

+  987.16 

+  1020.63 

762.76 

16, 

+  492.48 

+  224.40 

+  265.44 

-  270.28 

26, 

+  550.04 

+   (4.74) 

-  (45.88) 

POOE,  The  Action  of  Jupiter  upon  Comet  V,  1889. 


43 


& 

J 

/ 

oy 

+  3491.34 

Mch.  26, 

-  513.96 

-  12754.77 

—  12797.60 

+  2977.38 

Apr.  5, 

-  465.72 

9777.39 

—  9816.20 

+  2511.66 

15, 

-  443.28 

7265.73 

-  7302.67 

+  2068.38 

25, 

-  427.36 

5197.35 

5232.13 

+  1641.02 

May  5, 

-360.00 

-  3556.33 

—  3586.33 

+  1281.02 

15, 

-  332.24 

2275.31 

-  2302.83 

+  948.78 

25, 

-  290.40 

1326.53 

1350.73 

+  658.38 

June  4, 

-  247.80 

668.15 

-  688.80 

+  410.58 

14, 

-200.28 

-  257.57 

274.26 

+  210.30 

24, 

-  154.44 

47.27 

60.14 

+  55.86 

July  4, 

-  103.12 

+   (4.28) 

+   (8.59) 

0.00 

+  8.66 

July  4, 

-  14.94 

+  61.10 

+  59.85 

-  6.28 

8, 

-10.44 

+  54.82 

+  53.95 

-  16.72 

12, 

5.04 

+  38.10 

+  37.68 

-  21.76 

16, 

+  3.96 

+  16.34 

+  16.67 

-  17.80 

20, 

+  19.80 

1.46 

+  0.19 

+  2.00 

24, 

-  6.53 

-  (1.26) 

+  (0.54) 

0.00 

—  4.52 

28, 

-10.44 

-  3.98 

44  POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889. 


July  20, 

+  199.64 

24, 

-  99.24 

-  304.40 

-  312.67 

+  100.40 

28, 

-  21.84 

-204.00 

—  205.82 

+  78.56 

Aug.  1, 

-  16.92 

-  125.44 

-  126.85 

+  61.64 

5, 

-  18.52 

—  63.80 

-  65.34 

+  43.12 

9, 

-  20.50 

-  20.68 

-  22.39 

+  22.62 

13, 

—  22.52 

+  1.94 

+  0.06 

+  0.10 

17, 

—  24.50 

+  2.04 

0.00 

+  2093.48 

Aug.  17, 

-  237.72 

—  7263.19 

-  7293.00 

'  +  1855.76 

•*''  27, 

-  269.40 

-  5407.43 

-  5429.88 

+  1586.36 

Sept.  6, 

-  251.16 

-  3821.07 

-  3842.00 

+  1335.20 

16, 

-  262.92 

-  2485.87 

-  2507.78 

+  1072.28 

26, 

-  280.92 

-  1413.59 

-  1437.00 

+  791.36 

Oct.  6, 

-  307.44 

' 

622.23 

647.85 

4-  483.92 

16, 

-  317.64 

-  138.31 

164.78 

+  166.28 

26, 

—  335.60 

-   (1.52) 

+  (27.97) 

POOR,  The  Action  of  Jupiter  upon  Comet  F,  1889. 


45 


eft* 

J 

7 

dz 

-  2309.40 

Mch.  26, 

+  336.24 

+  8580.58 

+  8608.60 

-  1973.16 

Apr.  5, 

+  308.64 

+  6607.42 

+  6633.14 

-  1664.52 

15, 

+  290.52 

+  4942.90 

+  4967.11 

-  1374.00 

25, 

+  267.24 

+  3568.90 

+  3591.17 

-  1106.76 

May  5, 

+  234.12 

+  2462.14 

+  2481.65 

872.64 

15, 

+  220.25 

+  1589.50 

+  1607.85 

-  652.39 

25, 

+  195.24 

+  937.11 

+  953.38 

457.15 

June  4, 

+  165.00 

+  479.96 

+  493.71 

292.15 

14, 

+  139.56 

+  187.81 

+  199.44 

152.59 

24, 

+  110.88 

+  35.22 

+  44.46 

-  41.71 

July  4, 

-f  77.94 

-   (2.74) 

-   (6.49) 

0.00 

7.34 

July  4, 

+  11.10 

-  37.22 

-36.29 

+  3.76 

8, 

+  6.98 

-33.46 

-  32.87 

+  10.74 

12, 

+  3.42 

-  22.72 

-  22.43 

+  14.16 

16, 

-  5.82 

-  8.56 

9.04 

+  8.34 

20, 

8.16 

—  0.22 

-  0.90 

+  0.18 

24, 

+  0.50 

+  (0.43) 

-  (0.04) 

0.00 

+  0.68 

28, 

+  2.04 

+  0.64 

46  POOR,  The  Action  of  Jupiter  upon  Comet  V,  1889. 


July  20, 
24, 

28, 
Aug.  1, 

+  2.40 
+  2.52 
+  3.06 

-  22.02 
-  19.62 
-  17.10 

+  65.99 
+  46.37 
+  29.27 

+  66.19 
+  46.58 
+  29.52 

-  14.04 

5, 

+  3.84 

+  15.23 

+  15.65 

-  10.20 

9, 

+  4.67 

+  5.03 

+  5.42 

—  5.53 

13, 

+  5.50 

-  0.50 

-  0.04 

-  0.03 

17, 

+  6.32 

-  0.53 

0.00 

-664.15 

Aug.  17, 

+  70.92 

+  2463.65 

+  2469.56 

-593.23 

27, 

+  65.88 

+  1870.42 

+  1875.91 

-527.35 

Sept.  6, 

+  71.52 

+  1343.07 

+  1349.03 

—  455.83 

16, 

+  81.36 

+  887.24 

+  894.02 

-  374.47 

26, 

+  92.16 

+  512.77 

+  620.45 

-  282.31 

Oct.   6, 

+  104.04 

+  230.46 

+  239.13 

-  178.27 

16, 

+  115.44 

+  52.19 

+  61.81 

-  62.83 

26, 

+  127.71 

+  (1-02) 

-  (10.64) 

Fie/.  A. 


PLATE    I. 


Fig.  B. 


-rX 


Sept.  26 


_X 


2  4- 


- Sept.  26 


Sept..  17 


July  20 


...  June  24 


June 24. 


....Mch.26 
I  1* 

Oct.  26 

i 

RELATIVE  ORBIT 

Scale,  1"  =  40  radii  of  Jupiter. 

(See  page  27.) 


,~ M&y/s 


Mch.26 


ACTUAL  OHB/T 

Scale,  5"  =  mean  distance  of  ®. 


PLATE   II. 


..Saturn 


NEW  AND  OLD  ORBITS  OF  COMET  V,  1889. 

(See  page  31.) 


